Logarithms: Concepts & Theory

Logarithm is the inverse of exponential. To understand the concept of logarithm, let us start with an example. What is the value of 34?
Well!! We know that 34 = 81.
Now if you are asked the same question but in a different way like “what should be the exponent of 3 to get the result 81?”
The answer is 4. The above question is the basic definition of logarithms.
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Now the question is that how will you write the above question in the form of logarithm??
We will write it as log3 81 = ?
Here, 3 is the base whose exponent we are looking for. So we want to find the value which when rose as power to 3 will be equal to 81. Since such value is 4, so we can say that log3 81 = 4
Here the above equation will be read as “log base 3 of 81 is 4”.
So from here we can derive the general result as loga x = b ⇒ x = ab.
The above equation is the basic definition of the logarithm
Now let us learn some important log rules which will be used to solve the questions of logarithm. In some books, these logarithm formulas are also referred as basic log rules or rules for logarithms.
  • We know that a0 = 1. Hence we have loga 1 = 0
  • We have a1 = a. So we have loga a = 1
  • loga (x×y) = logax + logay
  • logax/y = logax-logay
  • loga xn = n × loga x
  • logax = logbx/logba , This result is called the change of base formula.
  • logax = 1/logxa , This is an another form of the change of base formula.
  • logab x = (1/b)logax
Following are some of the log properties which you must remember while solving the logarithms examples.
  • Logarithm of negative numbers and zero is not defined.
    Let us have loga x = b ⇒ x = ab. Now ab is always a positive number whatever be the values of a and b. So, x > 0 always. Hence ‘x’ cannot be negative or zero.
  • Base of a logarithm cannot be 1.
    Take an example like log1 3 = b ⇒ 3 = 1b. Now this can never be true. So the base cannot be equal to 1.
  • The base of a logarithm is always positive.
  • If no base is written then it is to be taken as 10.
You must remember all the above points related to log formulas to crack the questions on logs. In some books, these concepts are also defined and used in functions. The technical name referred to such questions or formulas is logarithmic functions.
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